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Elimination Without Eliminating: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets

Paul Breiding, John Cobb, Aviva K. Englander, Nayda Farnsworth, Jonathan D. Hauenstein, Oskar Henriksson, David K. Johnson, Jordy Lopez Garcia, Deepak Mundayur

TL;DR

The paper addresses computing the real regions in the complement of hypersurfaces that arise as projections, where obtaining an explicit defining polynomial is often infeasible. It introduces a framework of routing functions and gradient roadmaps for the known-$h$ case, and extends it to the unknown-$h$ setting using pseudo-witness sets to extract the necessary degree, gradient, and Hessian information from projections. The authors derive gradient/Hessian formulas for $\log|h|$ in terms of line-based intersections and monodromy, and present a complete algorithm that recovers the region structure without symbolic elimination, demonstrated across Kuramoto, 3RPR, and Allee-effect models. The work enables efficient, elimination-free analysis of discriminants and projected hypersurfaces, with practical impact in topology, dynamics, and mechanism design, and is implemented (forthcoming) in a Julia package.

Abstract

Many hypersurfaces in algebraic geometry, such as discriminants, arise as the projection of another variety. The real complement of such a hypersurface partitions its ambient space into open regions. In this paper, we propose a new method for computing these regions. Existing methods for computing regions require the explicit equation of the hypersurface as input. However, computing this equation by elimination can be computationally demanding or even infeasible. Our approach instead derives from univariate interpolation by computing the intersection of the hypersurface with a line. Such an intersection can be done using so-called pseudo-witness sets without computing a defining equation for the hypersurface - we perform elimination without actually eliminating. We implement our approach in a forthcoming Julia package and demonstrate, on several examples, that the resulting algorithm accurately recovers all regions of the real complement of a hypersurface.

Elimination Without Eliminating: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets

TL;DR

The paper addresses computing the real regions in the complement of hypersurfaces that arise as projections, where obtaining an explicit defining polynomial is often infeasible. It introduces a framework of routing functions and gradient roadmaps for the known- case, and extends it to the unknown- setting using pseudo-witness sets to extract the necessary degree, gradient, and Hessian information from projections. The authors derive gradient/Hessian formulas for in terms of line-based intersections and monodromy, and present a complete algorithm that recovers the region structure without symbolic elimination, demonstrated across Kuramoto, 3RPR, and Allee-effect models. The work enables efficient, elimination-free analysis of discriminants and projected hypersurfaces, with practical impact in topology, dynamics, and mechanism design, and is implemented (forthcoming) in a Julia package.

Abstract

Many hypersurfaces in algebraic geometry, such as discriminants, arise as the projection of another variety. The real complement of such a hypersurface partitions its ambient space into open regions. In this paper, we propose a new method for computing these regions. Existing methods for computing regions require the explicit equation of the hypersurface as input. However, computing this equation by elimination can be computationally demanding or even infeasible. Our approach instead derives from univariate interpolation by computing the intersection of the hypersurface with a line. Such an intersection can be done using so-called pseudo-witness sets without computing a defining equation for the hypersurface - we perform elimination without actually eliminating. We implement our approach in a forthcoming Julia package and demonstrate, on several examples, that the resulting algorithm accurately recovers all regions of the real complement of a hypersurface.
Paper Structure (15 sections, 4 theorems, 46 equations, 7 figures, 4 algorithms)

This paper contains 15 sections, 4 theorems, 46 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. The case of a known defining polynomial
  3. Routing points and routing functions on Rn
  4. Gradient flow and connectivity via routing functions
  5. Homotopy continuation and monodromy solving
  6. Summary of the algorithm for known defining polynomial
  7. The case of an unknown defining polynomial
  8. Witness and pseudo-witness sets
  9. Evaluating the gradient and the Hessian
  10. Summary of the algorithm for unknown defining polynomial
  11. Case studies
  12. Kuramoto model
  13. 3RPR
  14. Allee effect
  15. Conclusion

Key Result

Theorem 2.2

Let $r$ be as in eq:routing-function, with $e>\deg(h)/2$. Then, there exists a nonempty Zariski open subset $\mathcal{U}\subset\mathbb{R}^n$ such that for every $c=(c_1,\ldots,c_n)\in\mathcal{U}$, the following properties hold:

Figures (7)

  • Figure 1: The quadratic discriminant $\{h(a,b) = 0\}$ is the black curve in the picture. There are two regions, one above and one below the black curve. Our algorithm represents these regions by critical points (green) of a routing function (with level sets illustrated with colors between magenta and yellow). There is one critical point in the top region and three critical points in the bottom region. The latter three points are connected by gradient flow (the blue curves). The critical points and the flow trajectories were computed without direct access to the polynomial $h = a^2 -4b$.
  • Figure 2: A witness set for the red curve $X = V(F)$ with $F$ as in \ref{['ex:quadratic_discriminant2']}. The witness set is given by the intersection of the red curve $X$ with the green linear space $\mathcal{M}$. The intersection consists of the two points labeled $\mathcal{W}$.
  • Figure 3: The left picture illustrates a pseudo-witness set for the red curve $X = V(F)$. While \ref{['fig:WitnessQuad']} the green linear space $\mathcal{M}$ was general, here it must be a product space: $\mathcal{M} = \mathcal{L} \times \mathbb C$, where $\mathcal{L}$ is a general line; see \ref{['ex:quadratic_discriminant3']}. The pseudo-witness set is then given by the intersection of the red curve $X$ with $\mathcal{L} \times \mathbb C$. The projection of $\mathcal{W}$ onto the yellow two-dimensional plane yields ${\mathcal{H}} \cap \mathcal{L}$. The right picture shows the situation within the yellow plane.
  • Figure 4: Routing points and connecting paths for the discriminant of the two-parameter version of the Kuramoto model with three oscillators \ref{['kuramoto_eq']}. The left picture shows a zoomed out version of the right picture. The gradient flow is visible in blue. The arrows point towards index-zero routing points. We see that $7$ out of $9$ regions are inside the star-shaped figure. The last two regions are the inside of the ellipse and the unbounded region outside the ellipse. The pictures also display the level sets of the routing function $\log r$. The routing points and the flows were computed without access to the discriminant polynomial.
  • Figure 5: An example of a 3RPR mechanism.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Example 1.1
  • Definition 2.1
  • Theorem 2.2: SmoothConnectivity
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Remark 3.1
  • Definition 3.2
  • Example 3.3
  • Example 3.4
  • ...and 8 more